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# How old are these lentils?

January 27, 2017

Do you keep a jar of lentils? Lentils are those little lens-bean looking things that often mix so well with rice and other ingredients to make tasty and healthy meals. How old are the lentils in your jar?

Say you have a 1 gallon (3.785 litres) jar and it is filled to the top with lentils. Strictly speaking it is filled with lentils and air, but let's ignore the air and just consider the space between lentils as being an integral part of the lentils. Just like us humans, them lentils need some breathing space also.

How often do you use the jar? Say that every month you consume 1/2 a gallon. That is, half of the jar.

Say also that when your jar reaches half full, you go out and buy new lentils and refill your jar, mixing the old and new lentils. So what's in your jar? Old lentils? New lentils? Well, it is a mix. A mix of the new lentils you just bought and old lentils left over from last month.

But you have been keeping this lentil jar for years and it probably has lentils from the month before last month, and the month before that, and the month before that and so forth.

So how old is your oldest lentil?

At this point I urge you to either look in your pantry for your jar of lentils, or alternatively envision a jar of lentils? Think of these little friendly, lens looking beans in there. Is there a lentil that has been in the jar longer than any of its friend lentils? Or maybe there are a few lentils that are the "oldest gang", that is, they came into the jar together, quite a few months ago.

There is not a concrete answer to such lentil thoughts, and when searching the web for lentil jar experts, I didn't find any distinguished professor that has made a career, specifically based on lentil modelling. Nevertheless, if you let your mind veer off and think about lentil jars, you may enjoy discovering a few interesting things about mathematics. All in your mind - and maybe in the jar of lentils.

So think about the contents of our jar as it is at the day in which you buy new lentils, just after filling it up.

Q: How many of the lentils are new?

A: 1/2 of them.

Q: How many of the lentils are 1 month old?

A: ....(thinking).... These are the lentils that came in 1 month ago. Half of them came in one month ago. And since the jar is mixed up when putting in the lentils, about half of those were used last month and half remained. So a half of a half is 1/4.

Q: OK, so how many of the lentils are 2 months old?

A: Well, continuing with similar reasoning, 2 months ago we put in half. Then half of those were left after one month and half of that amount survived another month. So half of half of half is (1/2)x(1/2)x(1/2) = 1/8.

Q: So does this continue forever? That is, are 1/16 of the lentils 3 months old, 1/32 of the lentils 4 months old, 1/64 of the lentils are 5 months old etc...

A: Yes, perhaps so. Take lentils that were brought in N months ago. N here is a variable, for example N = 5 means 5 months ago. The amount brought in N month ago is 1/2 a gallon of lentils.  Since then, every month half of that was used and half remained. This happened for N months. Hence,

Wow, that is quite neat. But wait, say you are wondering how many lentils are 2.5 years old. A calculation using N=30 reveals that only about billionth of the jar's contents has lentils that old! But a billionth of a one gallon jar, is roughly about a millionth of a gram of lentils. That is maybe enough volume for a small biological cell, not a whole lentil! Can this be?

Well, our calculation assumed that lentils can be split up indefinitely and ignored the finer features of randomness. This would make some sense if we were talking about water, although in that case we also eventually end up with individual molecules. But for lentils, the calculation is great for N=1, N=2, and perhaps up to around N=10 or N=12. However after that, it is more theoretical than practical.

Nevertheless, the above thought experiment indicates that the following is probably true:

Here the three dots imply to continue adding smaller and small fractions forever!

Q: How come?

A: There is a total of 1 gallon in the jar. 1/2 of it is new, 1/4 is 1 months old, 1/8 is 2 months old and if this jar existed forever then the pattern must continue forever. No?

In mathematics, this infinite sum is called a convergent geometric series. You can sum up this sequence and never stop, and the more terms you add to the sum, the closer you reach 1. The sequences converges to 1. A non-lentil way to visualise such a series is to think of steps you are taking towards a wall. Initially 1 meter away, and with every step you move half way towards the wall. If you were as small as a lentil you could do quite a few steps until hitting the wall. If you were an infinitely small dot, you would do an infinite number of steps and get very close, but never touch it.

Ok... we digressed from lentils to walls and indeed, there is so much more that can come from thinking about a simple jar of lentils without hitting the wall. The next time you make a meal with your loved ones, perhaps drive their curiosity and nurture your connection with them by exploring such lentil matters. Let us know where the discussion goes.

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